PHYS 339: Physical Mechanics Fall 2020 Block C Kirkman
Text: Classical Mechanics by John R. Taylor (University Science Books 2005)
Class 1 (M Oct 26)
1.2, 1.4, 1.10, 2.18, 2.43, canA.pdf
Class 2 (T Oct 27)
old exam 339t1_17.pdf #4
5.8, 5.13
In the folder 2017 find the file of Mathematica commands
SHO1.m
which solves the differential equation for ten example damped oscillators (mostly driven)
and plots the ten resulting graphs of x vs t. The file
SHO1.pdf
shows the resulting plots. Report the value of omega0 (all ten have the same value);
Report the values of Q (the first eight have the same value). The first five solutions seem
to have immediately achieved the equilibrium state. If the system is started in the equilibrium
solution (Eq. 5.66) it will continue. Following Eq. 5.66, the initial position and velocity:
x0=A Cos[-delta]=A Cos[delta]
v0=Aw Sin[delta]
Thus:
A=Sqrt[x0^2+(v0/w)^2]
Tan[delta]=v0/(w x0)
In a spreadsheet (because you'll be making the same calculation five times), calculate A and delta
for these five solutions. Does A approximately mimic Figure 5.18? Does delta approximately
mimic Figure 5.19? Do the x vs t graphs look proper (i.e., if delta is small, x should look like
a Cos; if delta is pi/2 what should it look like and does it?; if delta is pi what should
it look like and does it? REMARK: the usual inverse tan function, atan, has a range -pi/2 to
+pi/2. The function atan2(x,y) has the full range -pi to pi, so atan2(-1,.01)=pi whereas
atan(y/x)=atan(.01/-1)=0. Use atan2 in your spreadsheet.
Annotate each of the ten plots: directly on the hardcopy adjacent to the appropriate
plot report the words that *describe* the oscillator (e.g., under damped, over damped, driven,
above resonance, below resonance etc) and describe how/why it *differs* from the neighboring plots.
If you have trouble understanding/explaining a plot you can copy and paste the code
into Mathematica and adjust the parameters to see what is controlling what.
You should have five values for omega & A from the first five plots. While five numbers
(none with errors) isn't a lot of points to fit to a curve, WAPP does have resonance curve
as a possible function. WAPP & fit; print out a plot. Does the Q match what you found earlier?
Directly on the plot display and measure what is usually described as the
"full width at half max" (but which is actually taken as max/sqrt(2)). Eq. 5.75
claims this should approximately equal 2*beta...does it?
Class 3 (R Oct 29)
old exam 339t1_17.pdf #1 add
(d) Calculate the total kinetic energy in both the pre-collision and post-collision states.
Calculate the kinetic energy "of the center of mass" in the post-collision state.
Remark: Since the velocity of the center of mass should be constant, you can use the
above to calculate the kinetic energy "about the CM" easily. However, I want you to
calculate the kinetic energy "about the CM" in the final state using the reduced mass formula.
Remark: the problem reports that the relative speed is unchanged; that means the
the kinetic energy "about the CM" is unchanged.
3.21, 3.22: the book says to use polar/spherical coordinates but I find these just as
easy in Cartesian so use what ever you want.
Find the moment of inertial of a square (side a) thin metal sheet about its CM
with the rotation axis perpendicular to the sheet.
To find I for objects with holes: you can subtract the I of the stuff removed to make the hole.
Using the subtraction idea, find the moment of inertia I of a thin spherical shell
(something like a balloon: thin film--say delta r thick--surrounding nothing).
In the end let delta r go to zero, but with the mass of the shell staying finite and
then express your final answer in terms of mass and radius of the shell.
Class 4 (F Oct 30)
old exam 339t1_17.pdf #3
problems: 4.2, 4.8 (see below), 4.12, 4.23
2017/4.8.pdf gives additional background on problem 4.8
Class 5 (M Nov 2) Exam 1
Class 6 (T Nov 3)
6.4, 6.9, 6.12
Class 7 (R Nov 5)
old exam 339t2_17.pdf #2, #3, #4, 7.14, 7.29, 7.40
Class 8 (F Nov 6)
chapter 8
also read: http://www.physics.csbsju.edu/orbit/orbit.2d.html
watch:orbit.webm
Homework (find pics in 2017 folder):
Mercury_Tracking_Network_2.png
shows the ground track of the 5th US manned space flight (Schirra, 3-Oct-1962).
The flight was launched from FL went about 6 orbits and finished (as planned)
with a splash-down in the Pacific. Note that the capsule did not fly over MN and
that the orbits 'advanced' a bit (e.g. orbit 3 crossed the equator at about 10 deg
and orbit 4 crossed at a bit less than 35 deg). Explain why these features
were natural consequences of orbital mechanics. *Calculate* (based on orbital mechanics)
the expected orbit advance and compare to the value visible in the ground track.
iss_track.jpg
shows some ground tracks of the International Space Station. This satellite
does sometimes pass over MN. Why the difference compared to Mercury5?
This image may help explain: iss.gif. The altitude of ISS for these tracks
was 370 km; calculate the period. (Note: the orbit is nearly circular.)
IRNSSschematic.jpg
shows some orbits of Indian Regional Navigational Satellite System.
Explain these orbital tracks.
Halley's comet has
eccentricity=e=.967
semi-major axis=a=17.8 AU (FYI: 1 AU = Earth-Sun semi major axis)
period=T=75.3 year
What fraction of the time is the comet more than its semi major from the Sun.
(Or more generally: if the eccentricity is e what fraction of the time is r>a?)
in 2017, for the first time, an asteroid that came from outside our Solar System was observed.
see wiki: Oumuamua
wiki reports (somewhat outdated) orbital data:
Perihelion 0.25534 AU
Semi-major axis -1.2798 AU (the negative sign just indicates hyperbolic; don't use it if you use my formulae)
Eccentricity 1.19951
At the time JPL calculated the (hyperbolic) orbit and reported:
Vinfinity=26 km/s
calculate this yourself!
"JPL solution indicates that one hundred years ago, the object was roughly 553 AU (83 billion km) from the Sun."
The above sentence refers to time, which occurs in only one formula I've given you:
e sinh(u)-u= omega t
if you knew u you could get r:
r=a(e cosh(u)-1)
but there is no way to "solve" the equation to get algebra for u as a function of t
I suggest FindRoot on mathematica or root finding on your calculator.
8.13, 8.19
Class 9 (M Nov 9)
chapter 13
hamiltonian20.pdf: #1,#2,#3
13.23
old exam: 339t2_17.pdf #1
these problems are algebraically messier than usual;
do note that my solution is posted and you are wlecome to consult it
Class 10 (T Nov 10)
exam 2; help?
Class 11 (R Nov 12)
read chapter 10
review the following videos from 2017/handouts/
IntermediateAxisTheorem.webm
dancingT.webm
rotating_in+microgravity.webm
from main page:
323_rotation_sequence_Euler.webm
Euler2a.gif
problems
10: 35,37
tetrahedron.pdf
old exam: 339t3_17.pdf #1, #2
consider the following three matrices: (Note A & B are 3x3; C is 3x2)
1 2 -1
A= 0 3 1
2 0 1
2 1 0
B= 0 -1 2
1 1 3
2 1
C= 4 3
1 0
calculate: det(A), det(B), det(A.B), A.C, transpose(C).A, inverse(B), trace(A.B)
report (calculation should not be needed): det(transpose(A)), det(inverse(B)), det(B.A), trace(B.A)
Feel free to use your calculator or Mathematica, but (perhaps except for the inverse) these are easy
to do "by hand".
Class 12 (F Nov 13)
10.43
The Earth's Chandler wobble is used in the textbook as an example of
free precession and as folks on Earth we are measuring in
the body-fixed frame. A Google search should come up with
a map showing the Chandler wobble over time: is the motion of
the pole (i.e., omega) in the direction of the Earth's rotation
or in the opposite direction? What direction is predicted by
Eq. 14 & 10.94? From space the Earth's symmetry axis would seem
to wobble. What would be the period of this motion?
Itensor.pdf
coin flip:
A coin in a horizontal plane is tossed into the air with angular velocity
omega1 about its diameter and omega3 about its symmetry axis.
IF omega3=0 the coin is flipping about its diameter; if omega1=0
the coin is simply spinning flat with the same face up.
What is the smallest value of omega3/omega1 for which the wobble is
such that the same face is always exposed to an overhead observer?
HINT: the fact that the coin starts horizontal is an important part of the problem.
HINT2: if in the intial part of the wobble the coin is horizontal, then the wobble axis cannot be vertical.
FYI: you can find lots of youtube slow motion clips of coin tosses, but those
I've found have omega3=0.
Why is it difficult to make a pencil (spinning on point) act like a precessing gyro?
Consider a typical pencil: length=15 cm; radius=.35 cm (the mass of
5 g is not needed as it will cancel). Treat the pencil as a solid
cylinder (moment of tensor for a solid cylinder about its CM is given
in Itensor.pdf; you need to use the parallel axis thm to figure I about an end)
Using the definitions in euler_angles+top.pdf and code in euler_angles+top.m
calculate "c" for the pencil-top. Run the code to find an "a" value
that shows something that looks like precession with minor (but visible) nutation.
Calculate the omega3 that is required for that "a" value.
Provide hardcopy of the locus-of-axis (ParametricPlot3D) and the NDSolve command used.
2017/handouts/B+O_7-25.pdf
Barger & Olsson (textbook authors) were professors at Madison when I was a student there.
The physics building had trash cans with lids matching this problem; they indeed made
fine gyroscopes. The only answer I seek is to "direction of precession": clockwise or counterclockwise
as viewed from above. Again: none of the other results are requested.
Class 13 (M Nov 16)
old exam: 339t3_17.pdf #3
In your exam2 #5 (3 homonuclear linear molecule) using the original x coordinates
find the mass matrix (should be easy), find the PE K matrix, find the normal mode frequencies
Find on this web site the spreadsheet diagonalizeI.xlsx
(It is much like the tetrahedron spreadsheet: you can input values for phi, theta, psi
and the resulting Euler matrix m & transpose(m)=mT are calculated for you.)
In the textbook we calculated the moment of inertia tensor for a cube using a vertex as origin:
8 -3 -3
I= -3 8 -3
-3 -3 8
(we're not bothering with the overall factor of Ma^2/12)
Plan: Find the matrix m such that after the similarity transform: m.I.mT
the resulting matrix is diagonal.
There is a theorem that any symmetric matrix can be diagonalized by an orthogonal matrix.
A lot of this problem is just learning how to do matrix multiplication in a spreadsheet.
in each of the 9 cells in box below I in the spreadsheet enter the formula that would calculate
the corresponding element of the matrix: m.I
in the 9 cells in the box below that box and using the results for m.I you just calculated
enter the formulas for the corresponding elements of m.I.mT The result is what the I
matrix would look like in the new coordinate system. Enter for the Euler angles:
phi=pi()/4, theta=atan(sqrt(2)), psi=0
This should diagonalize the I, i.e., we have transformed to the frame we've usually used
as our "body" frame.
You might recall that the eigenvalues of this matrix were: 11,11,2
Email me the resulting spreadsheet. (Note: if you're working within a group you can email
me one copy with all the names attached. As usual, make sure all the group members understand the solution!)
Note: use of the $ "hold fixed" sign on cell references can save a bit of time allowing carefully
constructed formulae to be "pulled down" or "pulled over" rather than recreated from scratch. Reminder:
$A1 will hold the reference to the first column fixed if the cell is copied (but the row will change from 1)
A$1 will hold the reference to the first row fixed if the cell is copied (but the column will change from A)
$A$1 will always refer to exactly the same cell if the cell is copied.
Class 14 (T Nov 17)
lagrange_points.pdf: complete *one* "Option" from the Lab section
For the Earth-Moon system,
old exam: 339t3_17.pdf #4-5 (feel free to use Mathematica)
9.11, 9.26, 9.28
Class 15 (R Nov 19)
final exam